Fractal relation of mainstream length to catchment area in river networks

Abstract

Mandelbrot's (1982) hypothesis that river length is fractal has been recently substantiated by Hjelmfelt (1988) using eight rivers in Missouri. The fractal dimension of river length, d , is derived here from the Horton's laws of network composition. This results in a simple function of stream length and stream area ratios, that is, d = max (1, 2 log R L /log R A ). Three case studies are reported showing this estimate to be coherent with measurements of d obtained from map analysis. The scaling properties of the network as a whole are also investigated, showing the fractal dimension of river network, D , to depend upon bifurcation and stream area ratios according to D = min (2, 2 log R B /log R A ). These results provide a linkage between quantitative analysis of drainage network composition and scaling properties of river networks.